Good snag on the 3 screw. I like them and yes get a good trigger job. From a good Smith.
You are fortunate in that you shoes the 44 mag. Most of the throats on 44 cylinders were fine. Measure them ,,,,
More accurate is with Mic or Pin gauges if you don't have these then get a good jacketed bullet and see if you can just press fit a bullet through the throat. Ruger was pretty good on getting the 44 cylinders correct. I scream shoot the gun first and see what is going on. I have a Ruger with tight throats that still shoots a 2" group at 34 yds, oh and it has a 3 1/2" barrel
You can check the bore with a kit from Beartooth bullets. They have the soft lead inserts for doing this. I call them inserts since they have a leg to fit on a cleaning rod. this helps. Once you know the bore size you can have a better understanding about you cylinder throats. Running the lead slug thru the barrel tells you also if there is any tight spots or loose ones, and if there is thread choke in the barrel. This is the area at which the barrel is threaded into the frame. If that slug wants to stop right there then you have thread choke, and two ways to get rid of it. Firelapping, or having a Smith do a Taylor throat on the barrel. Beartooth also sells a kit for "Firelapping" your barrel if there is thread choke.
As far as the forcing cone is concerned, Ruger was pretty good on this. No matter what degrees it is 5, 8, or 11, make sure that it is even and not damaged in anyway.
You had ask about tricks,,, these not necessarily being tricks but things that can aid accuracy or functionality.
Belt Mountain base Pin GET ONE! I prefer the locking base pins. I also like to get a stronger base pin latch spring. ( heavy loads ) this is better.
Sights,,, You might not have time right now but you might want to consider some different front sights. White line in the front sight, fiber optics, a patridge front sight,
Grips, I enjoy custom grips but this is going to be up to you as to what kind and what material.
Grip Frame. This goes hand in hand with the grips, especially if you go the custom route. You might need the frame decked ( skimmed on a Mill to make sure it is flat) The middle knuckle may get hit and or bruised and you need a little relief mods on the grip frame or even trigger guard. You will discover all this soon.
But for now, start shooting that gun and enjoy it. If you want to shoot milder loads as practice and get trigger time to work on grip and trigger control and to "learn" the gun.
Try some different loads Jacketed and cast if you want,,, just to see what the gun likes or dislikes.
You do not have to have the maximum power dynamite loads the 44 will do a good job with milder Fodder.
Barnes are great,, and Federal loads Swift bullets... About 20.00 a box less than Swifts own brand of ammo. Top choices for hunting. Hornady 240gr XTP is very accurate. Not my choice for hunting But that's my 2 cents.
But if you do already, or start rolling your own, then you can load some cast HPs, or solids or good jacketed, and have more to choose from, and tweak and tailor your loads to suit your gun.